Top 10 Arxiv Papers Today
in Representation Theory

0.0 Mikeys

Green's theorem states that the Hall algebra of the category of
representations of a quiver over a finite field is a twisted bialgebra.
Considering instead categories of orthogonal or symplectic quiver
representations leads to a class of Hall modules which are also comodules over
the Hall algebra. A module theoretic analogue of Green's theorem is not known.
In this paper we prove module theoretic analogues of Green's theorem in the
degenerate settings of finitary Hall modules of
$\mathsf{Rep}_{\mathbb{F}_1}(Q)$ and constructible Hall modules of
$\mathsf{Rep}_{\mathbb{C}}(Q)$. The result is that the module and comodule
structures satisfy a compatibility condition reminiscent of that of a
Yetter--Drinfeld module.

Github

Youtube

Other stats

0.0 Mikeys

Elie Casbi

We study a compatibility relationship between Qin's dominance order on a
cluster algebra $\mathcal{A}$ and partial orderings arising from
classifications of simple objects in a monoidal categorification $\mathcal{C}$
of $\mathcal{A}$. Our motivating example is Hernandez-Leclerc's monoidal
categorification using representations of quantum affine algebras. In the
framework of Kang-Kashiwara-Kim-Oh's monoidal categorification via
representations of quiver Hecke algebras, we focus on the case of the category
$R-gmod$ for a symmetric finite type $A$ quiver Hecke algebra using
Kleshchev-Ram's classification of irreducible finite dimensional
representations.

Github

Youtube

Other stats

0.0 Mikeys

Claus Michael Ringel

Text of a (pre-dinner) lecture at the Bielefeld workshop "Discrete Categories
in Representation Theory", April 20 - 21, 2018. This workshop was organized in
order to celebrate the 60th birthday of Dieter Vossieck: his famous paper "The
algebras with discrete derived category" has to be seen as the starting point
of a development which is discussed in this workshop. Dieter Vossieck has
published only few papers, but his influence is much larger. We outline some of
these contributions.

Github

Youtube

Other stats

0.0 Mikeys

Hiroyoshi Tamori

Based on an idea in [Gan--Savin, Represent. Theory (2005)], we give a
classification of minimal representations of connected simple real Lie groups
not of type $A$. Actually, we prove that there exist no new minimal
representations up to infinitesimal equivalence.

Youtube

Other stats

0.0 Mikeys

Maxime Pelletier

In this article we study, in the context of complex representations of
symmetric groups, some aspects of the Heisenberg product, introduced by Marcelo
Aguiar, Walter Ferrer Santos, and Walter Moreira in 2017. When applied to
irreducible representations, this product gives rise to the Aguiar
coefficients. We prove that these coefficients are in fact also branching
coefficients for representations of connected complex reductive groups. This
allows to use geometric methods already developped in a previous article,
notably based on notions from Geometric Invariant Theory, and to obtain some
stability results on Aguiar coefficients, generalising some of the results
concerning them given by Li Ying.

Github

Youtube

Other stats

0.0 Mikeys

Yu Liu,
Panyue Zhou

For a triangulated category T, if C is a cluster-tilting subcategory of T,
then the quotient category T\C is an abelian category. Under certain
conditions, the converse also holds. This is an very important result of
cluster-tilting theory, due to Koenig-Zhu and Beligiannis.
Now let B be a suitable extriangulated category, which is a simultaneous
generalization of triangulated categories and exact categories. We introduce
the notion of pre-cluster tilting subcategory C of B, which is a generalization
of cluster tilting subcategory. We show that C is cluster tilting if and only
if B/C is abelian.

Github

Youtube

Other stats

0.0 Mikeys

Siddhartha Sahi,
Hadi Salmasian,
Vera Serganova

For a finite dimensional unital complex simple Jordan superalgebra $J$, the
Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak
g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak
g_\flat(1)$, such that $\mathfrak g_\flat(-1)\cong J$. Set $V:=\mathfrak
g_\flat(-1)^*$ and $\mathfrak g:=\mathfrak g_\flat(0)$.
In most cases, the space $\mathcal P(V)$ of superpolynomials on $V$ is a
completely reducible and multiplicity-free representation of $\mathfrak g$,
with a decomposition $\mathcal P(V):=\bigoplus_{\lambda\in\Omega}V_\lambda$,
where $\left(V_\lambda\right)_{\lambda\in\Omega}$ is a family of irreducible
$\mathfrak g$-modules parametrized by a set of partitions $\Omega$. In these
cases, one can define a natural basis
$\left(D_\lambda\right)_{\lambda\in\Omega}$ of "Capelli operators" for the
algebra $\mathcal{PD}(V)^{\mathfrak g}$. In this paper we complete the solution
to the Capelli eigenvalue problem, which is to determine the scalar
$c_\mu(\lambda)$ by which $D_\mu$ acts on...

Figures

Tweets

Github

Youtube

Other stats

0.0 Mikeys

Hassain M,
Pooja Singla

Let $\mathfrak{o}$ be the ring of integers of a non-Archimedean local field
$F$ with finite residue field of even characteristic. We prove that the
abscissa of convergence of representation zeta function of Special Linear group
$\mathrm{SL}_2(\mathfrak{o})$ is $1.$ In comparison of this result with respect
to group algebras, we prove that for compact discrete valuation rings
$\mathfrak{o}$ and $\mathfrak{o}'$ such that $\mathfrak{o}/\wp \cong
\mathfrak{o}'/\wp'$, $2 \mid |\mathfrak{o}/\wp|$, and
$\mathrm{Char}(\mathfrak{o}) \neq \mathrm{Char}(\mathfrak{o}')$ the complex
group algebras $\mathbb C[\mathrm{SL}_2(\mathfrak{o}/ \wp^{2k})]$ and $\mathbb
C[\mathrm{SL}_2(\mathfrak{o}'/ (\wp')^{2k})]$ are not isomorphic for any $k
\geq 2$. We also include a complete description of the primitive irreducible
representations of groups $\mathrm{SL}_2(\mathbb Z/2^{2k} \mathbb Z)$ and of
$\mathrm{SL}_2(\mathbb F_2[t]/(t^{2k}))$ for $1 \leq k \leq 3 $.

0.0 Mikeys

In this paper we investigate locally free representations of a quiver Q over
a commutative Frobenius algebra R by arithmetic Fourier transform. When the
base field is finite we prove that the number of isomorphism classes of
absolutely indecomposable locally free representations of fixed rank is
independent of the orientation of Q. We also prove that the number of
isomorphism classes of locally free absolutely indecomposable representations
of the preprojective algebra of Q over R equals the number of isomorphism
classes of locally free absolutely indecomposable representations of Q over
R[t]/(t^2). Using these results together with results of Geiss, Leclerc and
Schroer we give, when k is algebraically closed, a classification of pairs
(Q,R) such that the set of isomorphism classes of indecomposable locally free
representations of Q over R is finite. Finally, when the representation is free
of rank 1 at each vertex of Q, we study the function that counts the number of
isomorphism classes of absolutely indecomposable locally free...

Github

Youtube

Other stats

0.0 Mikeys

Jessica Fintzen

Let k be a non-archimedean local field with residual characteristic p. Let G
be a connected reductive group over k that splits over a tamely ramified field
extension of k. Suppose p does not divide the order of the Weyl group of G.
Then we show that every smooth irreducible complex representation of G(k)
contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that
every irreducible supercuspidal representation arises from Yu's construction.
This improves an earlier result of Kim, which held only in characteristic
zero and with a very large and ineffective bound on p. By contrast, our bound
on p is explicit and tight, and our result holds in positive characteristic as
well. Moreover, our approach is more explicit in extracting an input for Yu's
construction from a given representation.